# The tessellation problem of quantum walks

## TL;DR

This paper investigates the tessellation problem in staggered quantum walks, establishing bounds for certain graph classes and proving the NP-completeness of deciding tessellability for k≥3.

## Contribution

It introduces a relationship between the chromatic number of the clique graph and tessellation cover size, and proves the NP-completeness of the tessellability decision problem.

## Key findings

- Relates chromatic number to tessellation cover size for specific graph classes.
- Proves NP-completeness of deciding k-tessellability for k≥3.
- Identifies classes of graphs with minimal tessellation covers.

## Abstract

Quantum walks have received a great deal of attention recently because they can be used to develop new quantum algorithms and to simulate interesting quantum systems. In this work, we focus on a model called staggered quantum walk, which employs advanced ideas of graph theory and has the advantage of including the most important instances of other discrete-time models. The evolution operator of the staggered model is obtained from a tessellation cover, which is defined in terms of a set of partitions of the graph into cliques. It is important to establish the minimum number of tessellations required in a tessellation cover, and what classes of graphs admit a small number of tessellations. We describe two main results: (1) infinite classes of graphs where we relate the chromatic number of the clique graph to the minimum number of tessellations required in a tessellation cover, and (2) the problem of deciding whether a graph is $k$-tessellable for $k\ge 3$ is NP-complete.